WARNING: Some of these instructions are potentially harmful and can cause data loss, even if they are followed correctly. Use at your own risk. If the VM is running and it has critical data that has not been backed up, use another method (e.g. Windows Server Backup, manual copy, etc. from within the VM) to back it up somewhere else before following these instructions. You've been warned.

Problem: Backup runs normally for one run, then fails because the checkpoints can't be created after that. The first backup left behind checkpoints that are halfway merged.

Cause: The default permissions that Hyper-V assigns to the checkpoint files aren't sufficient for the checkpoint service to delete the old files, so it crashes after completing the merge on the first VHDX file. (Root cause: when setting up the folder structure for the VM storage in a non-default location, Hyper-V isn't smart enough to adjust permissions on the folders.)

Good solution: Set the Hyper-V service GUID "account" to have FULL ACCESS rights to the folders where the VHDX files are stored. (It should have full access to the entire VM.) In PowerShell started with Run As Administrator:

First, get the GUID of the VMs on your Hyper-V server. (You can also get these from the permissions on the "stuck" AVHDX files.)

Bad solution: If you're in a hurry, like I was, just set the VM folders' permissions to grant FULL ACCESS to "Everyone". Then set the permissions as above after everything's fixed up.

To fix the broken checkpoint merges:

First, set the permissions as detailed above. Then shut down the virtual machine. If you're lucky, the merge should begin after a moment and complete successfully. It might go by quickly if the AVHDX's aren't very large. Check the folder to see if the checkpoint files are gone. If they are, the VM is good to go.

If the merge doesn't happen automatically, or seems to get stuck partway through, start up the VM again. If the merge still doesn't complete on its own, you'll have to merge it manually.

CAUTION: IF THERE IS MORE THAN ONE LEVEL OF CHECKPOINT, YOU NEED TO MERGE THE CHECKPOINT FILES IN THE CORRECT ORDER OR YOU *WILL* LOSE DATA!

Find out which checkpoints are the Parent of the others by using the "Inspect" option on the virtual disk in the VM's settings in Hyper-V. You'll need to start with the last checkpoint file--that's likely the one that's showing as the active virtual disk.

Example (merge one checkpoint into another and then into the main VM file):

If this gives you a "file in use" error, check Hyper-V again, it might have started a merge on its own--let it run.
If this gives you a permissions error, check that the files have the correct permissions, as detailed above, plus the Administrator rights.
If this gives you some other error, sorry, that's outside the scope of this document.

After they're merged, go back into Hyper-V, remove the entire virtual disk (if you try to replace the existing one, it still thinks there's a checkpoint pending). Click "Continue" on the warning about checkpoints. Re-add the base disk to the VM. Repeat for any additional virtual disks on that VM.

After starting up the VM again, you may have to put the second (third, fourth, etc.) disks online manually, within the VM, using Administration Tools -> Computer Management -> Disk Management. Everything should be back to normal after that.

Note: I didn't bother opening a Veeam support case because this issue wasn't specific to Veeam, but with Hyper-V, though it affects Veeam and other backup software that uses Hyper-V checkpoints, so figured I'd post the solution here.